Security control method of cyber-physical system under non-periodic denial of service attacks

Abstract

A safety control method for an information-physical system under a non-periodic denial-of-service attack belongs to the field of information-physical systems. The present invention solves the problem of poor control performance of current information-physical system control methods under non-periodic denial-of-service attacks. The present invention first provides sufficient conditions for a closed-loop information-physical system to achieve asymptotic stability under a non-periodic denial-of-service attack; secondly, within a time-varying attack cycle, the relationship between the attack activation interval and the attack silent interval is derived; then, sufficient conditions for evaluating the safety of a closed-loop information-physical system are proposed and a safety controller is designed; finally, a wheeled mobile robot is used to verify the effectiveness of the method of the present invention. The method of the present invention can be applied to the safety analysis and control of information-physical systems.

Description

Technical Field

[0001] The present invention belongs to the field of information-physical systems, and in particular relates to a security control method for information-physical systems under non-periodic denial-of-service attacks. Background Art

[0002] Cyber-physical systems, which deeply integrate advanced technologies such as perception, computing, communication, and automatic control, have gained great attention due to their reliability, intelligence, convenience, and scalability. Cyber-physical systems have been applied in various fields, such as agricultural production, aerospace, civil infrastructure, smart grids, power systems, and so on. Cyber-physical systems involve interactions between the network layer and the physical layer, which makes cyber-physical systems very vulnerable to attacks by malicious adversaries.

[0003] Compared with periodic denial of service attacks, non-periodic denial of service attacks show irregular time intervals and forms. Attackers can flexibly adjust the timing and method of attacks to cope with changes in defense measures. Therefore, for attackers, non-periodic denial of service attacks have greater practical value than periodic denial of service attacks. Under non-periodic denial of service attacks, the security measures taken by the information-physical system against periodic denial of service attacks will fail, and the control performance of the current control methods for information-physical systems against non-periodic denial of service attacks is also poor, or even ineffective. Therefore, under non-periodic denial of service attacks, the control performance obtained by the current control methods for information-physical systems is poor, and it is of great significance to study the security control strategy of information-physical systems under non-periodic denial of service attacks. Summary of the invention

[0004] The purpose of the present invention is to solve the problem of poor control performance of current information-physical system control methods under non-periodic denial of service attacks, and to propose a security control method for information-physical systems under non-periodic denial of service attacks.

[0005] The technical solution adopted by the present invention to solve the above technical problems is:

[0006] A security control method for a cyber-physical system under a non-periodic denial of service attack, the method specifically comprising the following steps:

[0007] Step 1: Build a cyber-physical system model;

[0008] Step 2: Based on the cyber-physical system model and the aperiodic denial of service attack model, a closed-loop system under aperiodic denial of service attack is constructed;

[0009] The non-periodic denial of service attack model is as follows:

[0010]

[0011] in, It is a non-periodic denial of service attack, using the collection represents the set of time-varying periods of aperiodic denial-of-service attacks, is the time-varying interval where there is no denial of service attack,

[0012]

[0013] Step 3: Construct the asymptotic stability condition of the closed-loop system under non-periodic denial of service attacks;

[0014] Step 4: Design a controller for the cyber-physical system under non-periodic denial-of-service attacks to ensure that the closed-loop system is asymptotically stable under non-periodic denial-of-service attacks.

[0015] The beneficial effects of the present invention are:

[0016] The present invention provides sufficient conditions for a closed-loop information-physical system to achieve asymptotic stability under non-periodic denial of service attacks, and then based on the given sufficient conditions for asymptotic stability, proposes sufficient conditions for evaluating the security of a closed-loop information-physical system, and designs a security controller for the information-physical system under non-periodic denial of service attacks, thereby improving the control performance of the information-physical system under non-periodic denial of service attacks.

[0017] Furthermore, the method of the present invention is applicable both in the presence and absence of external disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a block diagram of the cyber-physical system under a non-periodic denial of service attack;

[0019] In the figure, the controller and the actuator complete information exchange through the network layer. In order to prevent the transmission of control signals, the attacker can implement malicious non-periodic denial of service attacks;

[0020] Figure 2 are the actual trajectory and reference trajectory of the wheeled robot when Theorem 1 holds under different initial conditions;

[0021] Figure 3 is the actual trajectory and reference trajectory of the wheeled robot when Theorem 1 is not satisfied;

[0022] Figure 4 are the actual trajectory and reference trajectory of the wheeled robot under different initial conditions when Theorem 3 holds;

[0023] Figure 5 are the actual trajectory and reference trajectory of the wheeled robot when Theorem 3 is not satisfied. DETAILED DESCRIPTION

[0024] Specific implementation method 1: Combination Figure 1 The security control method of the cyber-physical system under the non-periodic denial of service attack described in this embodiment specifically comprises the following steps:

[0025] Step 1: Build a cyber-physical system model;

[0026] Step 2: Based on the cyber-physical system model and the aperiodic denial of service attack model, a closed-loop system under aperiodic denial of service attack is constructed;

[0027] from Figure 1 It can be seen that the controller-actuator communication network is blocked by malicious aperiodic denial of service attacks. The aperiodic denial of service attack model is as follows:

[0028]

[0029] in, It is a non-periodic denial of service attack, using the collection represents the set of time-varying periods of aperiodic denial-of-service attacks, is the time-varying interval where there is no denial of service attack,

[0030] Step 3: Construct the asymptotic stability condition of the closed-loop system under non-periodic denial of service attacks;

[0031] Step 4: Design a controller for the cyber-physical system under non-periodic denial-of-service attacks to ensure that the closed-loop system is asymptotically stable under non-periodic denial-of-service attacks.

[0032] Specific implementation method 2: This implementation method is different from the specific implementation method 1 in that when there is no external disturbance, the cyber-physical system model is as follows:

[0033] x k+1 =Ax k +Bu k (1)

[0034] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, and matrices A and B are constant matrices.

[0035] The matrix pair (A,B) is controllable, the state x kis testable and, beyond that, imposes no constraints on the system model.

[0036] The other steps and parameters are the same as those in the first embodiment.

[0037] Specific implementation method three: This implementation method is different from the specific implementation method two in that the specific process of step two is as follows:

[0038] When the attacker does not impose a non-periodic denial of service attack, the designed controller for:

[0039]

[0040] Where K represents the controller gain matrix to be designed;

[0041] Under a non-periodic denial of service attack, the control signal u received by the actuator is k for:

[0042]

[0043] Then the closed-loop system under the non-periodic denial of service attack is:

[0044]

[0045] The other steps and parameters are the same as those in the second specific implementation manner.

[0046] Specific implementation method 4: This implementation method is different from specific implementation method 3 in that the asymptotic stability condition is:

[0047] Given a scalar constant -1<σ a <0,σ b >0, if there exists a symmetric matrix P>0 and the gain matrix K satisfies equations (5) to (8), then the closed-loop system of equation (4) is asymptotically stable:

[0048] (A-BK) T P(A-BK)-P-σ a P<0 (5)

[0049] A T PA-P-σ b P<0 (6)

[0050]

[0051]

[0052] The superscript T represents the transpose of the matrix, and sup represents t i+1 -τ i The upper bound of , ln is the natural logarithm.

[0053] The other steps and parameters are the same as those in the third specific implementation manner.

[0054] Equations (7) and (8) are the relationship between the attack activation interval and the attack silent interval in a time-varying attack cycle. If the relationship between Equations (7) and (8) is not satisfied, the stability of the closed-loop system will deteriorate. Based on this, the security of the cyber-physical system can be effectively evaluated.

[0055] Specific implementation method 5: This implementation method is different from the specific implementation method 4 in that the specific process of step 4 is as follows:

[0056] Solve the matrix X that satisfies inequalities (9) and (10) p and matrix K p , and X p >0:

[0057]

[0058]

[0059] Wherein, the “*” in formula (9) represents (AX p -BK p ) T ; The “*” in formula (10) represents (AX p ) T , -1<σ a <0,σ b >0;

[0060] Then the gain matrix K is:

[0061]

[0062] in, is the matrix X p The inverse matrix of

[0063] Then, according to the gain matrix K, the controller of the cyber-physical system when there is no external disturbance is obtained:

[0064] The other steps and parameters are the same as those in the fourth embodiment.

[0065] In this embodiment, there is X p >0 and K p When inequalities (9) and (10) are satisfied, and equations (7) and (8) are satisfied, the closed-loop system of equation (4) can achieve asymptotic stability.

[0066] Specific implementation method 6: This implementation method is different from the specific implementation method 1 in that when there is an external disturbance, the cyber-physical system model under the external disturbance is:

[0067]

[0068] Among them, x k+1 is the system state at time k+1, x k is the system state at time k, Status x k It is measurable, k is the control signal received by the actuator at time k, A and B are constant matrices, and the matrix pair (A, B) is controllable. C, E, and F are known constant matrices. k represents the external disturbance at time k, and ω k belongs to l2[0,∞), l2[0,∞) represents the set of square integrable functions on the domain from 0 to positive infinity, z k is the control output at time k, is a real number, and m, n, p, and q are dimensions.

[0069] The other steps and parameters are the same as those in the first embodiment.

[0070] Specific implementation method 7: This implementation method is different from specific implementation method 6 in that the specific process of step 2 is as follows:

[0071] When the attacker does not impose a non-periodic denial of service attack, the designed controller for:

[0072]

[0073] Where K represents the controller gain matrix to be designed;

[0074] Under a non-periodic denial of service attack, the control signal u received by the actuator is k for:

[0075]

[0076] When there is a non-periodic denial of service attack, the closed-loop system under external disturbance is:

[0077]

[0078] The other steps and parameters are the same as those in the sixth embodiment.

[0079] Specific implementation eight: This implementation is different from specific implementation seven in that the asymptotic stability condition in step three is:

[0080] Given parameters When the external disturbance ω k = 0, the controller of formula (14) is designed to make the closed-loop system of formula (15) asymptotically stable; when the initial condition is 0, the control output z k satisfy

[0081]

[0082] If the above two conditions are met, the closed-loop system of equation (15) is asymptotically stable and has a guaranteed H ∞ Performance Level Next, the asymptotic stability conditions are given based on the above two conditions:

[0083] Given a scalar constant -1<σ a <0,σ b >0 and H ∞ Performance Level If there exists a symmetric matrix P>0 and a gain matrix K that satisfies the inequalities of equations (17) to (20), then the closed-loop system of equation (15) is asymptotically stable:

[0084]

[0085]

[0086]

[0087]

[0088] Among them, γ 11 =(A-BK) T P(A-BK)-P-σ a P+C T C,γ 21 =A T PA-P-σ b P+C T C, the superscript T represents the transpose of the matrix, I is the identity matrix, and sup represents t i+1 -τ i The upper bound of , ln is the natural logarithm.

[0089] The other steps and parameters are the same as those in the seventh embodiment.

[0090] Specific implementation method 9: This implementation method is different from specific implementation method 8 in that the specific process of step 4 is as follows:

[0091] Solve the matrix X that satisfies inequalities (21) and (22) p and matrix K p , and X p >0:

[0092]

[0093]

[0094] Among them, -1<σ a <0 and σ b >0;

[0095] Then the gain matrix K is designed as:

[0096]

[0097] Among them, the superscript -1 represents the inverse of the matrix;

[0098] Then, according to the gain matrix K, the controller of the cyber-physical system with external disturbance is obtained:

[0099] The other steps and parameters are the same as those in the eighth embodiment.

[0100] In this embodiment, there exists a matrix X that satisfies inequality (21) and inequality (22): p and matrix K p , and satisfy equations (19) and (20), the closed-loop system satisfies the asymptotic stability condition and has a guaranteed H ∞ Performance Level Right now

[0101] The present invention is described in detail below. The method of the present invention is mainly divided into the following steps:

[0102] (1) Description of the physical controlled object of the cyber-physical system without external disturbances

[0103] The physical controlled object of the cyber-physical system without external disturbance is described by the following linear discrete-time system:

[0104] x k+1 =Ax k +Bu k (twenty four)

[0105] in is the system status, is the control input received by the actuator, A, B are constant matrices with appropriate dimensions. The matrix pair (A, B) is controllable, and the state x k Other than that, no constraints are imposed on the system model.

[0106] For system (24), when the attacker does not impose a denial of service attack, the controller can be described as:

[0107]

[0108] Where K represents the controller gain value to be designed. Figure 1 It can be seen that the controller-actuator communication network is blocked by malicious aperiodic denial-of-service attacks. These attacks are described by the following periodic model

[0109]

[0110] in represents an indicator and represents the set of time-varying periods of denial of service attacks, is the time-varying interval where there is no denial of service attack, ι=1,2,…. The control instant is 0=t1<t2<…<t ι <…, and lim ι→∞ t ι →∞.

[0111] Under the non-periodic denial of service attack, the control signal received by the actuator is

[0112]

[0113] We can further obtain the following closed-loop system:

[0114]

[0115] (2) Security and stability analysis of cyber-physical systems under non-periodic denial-of-service attacks

[0116] Theorem 1 Given a scalar constant -1<σ a <0,σ b >0, if there exists a symmetric matrix P>0 and the gain matrix K satisfies the following inequality, then the closed-loop system (27) can achieve asymptotic stability.

[0117] (A-BK) T P(A-BK)-P-σ a P<0 (28)

[0118] A T PA-P-σ b P<0 (29)

[0119]

[0120]

[0121] Therefore, within a time-varying attack cycle, the relationship between the attack activation interval and the attack silent interval (30) and (31) is derived. If this relationship is not satisfied, the stability of the system will deteriorate. Based on this, the security of the cyber-physical system can be effectively evaluated.

[0122] (3) Safety controller design

[0123] Next, the controller gain K in (26) will be designed to ensure that the closed-loop system (27) is asymptotically stable.

[0124] Theorem 2 Given a scalar constant -1<σ a <0,σ b > 0, if conditions (30) and (31) are met, and there is X p >0 and K p If the following inequality is satisfied, the closed-loop system (27) can achieve asymptotic stability.

[0125]

[0126]

[0127] And the gain matrix K in (26) is as follows:

[0128]

[0129] (4) Safety analysis and control of cyber-physical systems under external disturbances

[0130] The dynamic system under external disturbance can be described as

[0131]

[0132] Where C, F and E are known constant matrices, represents the external disturbance, and ω k Belongs to l2[0,∞). Indicates control output.

[0133] Combining (26) and (35) we can get the following closed-loop system:

[0134]

[0135] The problem that needs to be solved now is as follows: given the parameters When the external disturbance ω k = 0, design the controller (26) so that the closed-loop system (36) achieves asymptotic stability; when the initial condition is 0, the control output z k satisfy

[0136]

[0137] If the above two conditions are met, then the system (36) is asymptotically stable and has a guaranteed H ∞ Performance Level Next, we will solve the above two problems. The following theorem gives the stability conditions.

[0138] Theorem 3 Given a scalar constant -1<σ a <0,σ b >0 and If there exists a symmetric matrix P>0 and a gain matrix K that satisfies the following inequality, then the closed-loop system (36) can be asymptotically stable and has a guaranteed H ∞ Performance Level

[0139]

[0140]

[0141]

[0142]

[0143] Among them, γ 11 =(A-BK) T P(A-BK)-P-σ a P+C T C,γ 21 =A T PA-P-σ b P+C T C.

[0144] (4) Design of security controller for cyber-physical systems under external disturbances and non-periodic denial-of-service attacks

[0145] Theorem 4 Given a scalar constant -1<σ a <0 and σ b > 0, if conditions (40) and (41) are met and there exists a matrix X p >0 and K p If the following inequality is satisfied, the closed-loop system (26) can be asymptotically stable and has a guaranteed H ∞ Performance Level

[0146]

[0147]

[0148] Furthermore, the controller gain K in equation (26) can be designed as follows:

[0149]

[0150] Experimental Section

[0151] In order to verify and demonstrate the effectiveness of the method proposed in this invention, a wheeled mobile robot will be used to illustrate the effectiveness of the control algorithm proposed in this invention. The motion of the mobile robot is determined by the vector [X, Y, θ] T It is represented by (X, Y) where (X, Y) represents the position coordinates of the mobile robot and θ represents the heading angle of the mobile robot. The units of X and Y are both m, and the unit of θ is rad.

[0152] The kinematic model of the mobile robot is as follows:

[0153]

[0154] Among them, v and Denote the linear velocity and angular velocity of the mobile robot respectively. The goal of the mobile robot is to track a circular trajectory, which is generated by the following virtual robot:

[0155]

[0156] Among them, [X r ,Y r ,θ r ] is the position and heading angle of the reference robot, v r and denote the linear velocity and angular velocity of the reference robot, respectively.

[0157] According to equations (45) and (46), the tracking error system is as follows:

[0158]

[0159] where e x =(YY r )sinθ+(XX r )cosθ,e y =(YY r )cosθ-(XX r )sinθ,e θ =θ-θ r .

[0160] Next, the linear discrete time model of the tracking error system of equation (47) is given:

[0161]

[0162] Where T is the sampling period.

[0163] According to formula (48), the physical controlled object model x of the cyber-physical system when there is no external disturbance is obtained: k+1 =Ax k +Bu k and the physical controlled object model of the cyber-physical system in the presence of external disturbances Among them, the system state x k It consists of the position coordinates (X, Y) of the mobile robot at time k and the heading angle θ of the mobile robot at time k.

[0164] (1) Safety analysis and controller design of cyber-physical systems without external disturbances

[0165] Two cases are given below to verify the effectiveness of the method proposed in the present invention:

[0166] Case 1: Tracking control of mobile robot when Theorem 1 is satisfied

[0167] Let T = 0.05, σ a =-0.1,σ b =0.7, select the sequence [t ι ,τ ι ) and [τ ι ,t ι+1 )as follows:

[0168]

[0169] It is easy to verify that conditions (30) and (31) in Theorem 1 hold. By solving the linear matrix inequality in Theorem 2, we can obtain the desired controller gain K and matrix X in equation (25): p as follows:

[0170]

[0171]

[0172] Select the desired position and heading angle as The desired linear and angular velocities are Figure 2 The robot's motion trajectories under four different initial conditions are shown. Figure 2 It can be seen that when Theorem 1 holds true, the control strategy proposed in the present invention can maintain the tracking performance of the robot.

[0173] Case 2: Tracking control of mobile robot when Theorem 1 is not satisfied

[0174] Now select σ a =-0.65,σ b =8.5, select the sequence [t ι ,τ ι) and [τ ι ,t ι+1 )as follows:

[0175]

[0176] Given the initial value [X, Y, θ] = [-2, -0.9, 0.2], the values ​​of other parameters are consistent with case 1. Through calculation, it can be seen that conditions (30) and (31) in Theorem 1 are no longer valid. Figure 3 shows the robot's motion, Figure 3 It can be seen that when the conditions derived from Theorem 1 are not met, the designed controller cannot maintain good tracking performance.

[0177] (2) Safety analysis and controller design of cyber-physical systems under external disturbances

[0178] The following two cases are provided to further verify the validity of the theories proposed in Theorem 3 and Theorem 4.

[0179] Case 1: Tracking control of mobile robot when Theorem 3 is satisfied

[0180] Select sampling period T = 0.1, parameter σ a =-0.2,σ b =1.37, Matrix C = [-2.6, -0.9, 0.1], matrix E = [0.1 0.2 0]. F and external disturbance signals are as follows:

[0181]

[0182] Select the sequence [t ι ,τ ι ) and [τ ι ,t ι+1 )as follows:

[0183]

[0184] By calculation, we can see that conditions (40) and (41) in Theorem 3 are established. By solving the linear matrix inequality in Theorem 4, we can obtain the desired controller gain K and matrix X in (25): p as follows:

[0185]

[0186]

[0187] The robot's expected position, heading angle, linear velocity, and angular velocity are consistent with the safety analysis of the cyber-physical system without external disturbances and the controller design. Under different initial conditions, the tracking performance of the controller designed by Theorem 4 is as follows: Figure 4 It is obvious that when conditions (40) and (41) are met, the controller designed by the present invention can maintain the desired tracking performance.

[0188] Case 2: Tracking control of mobile robot when Theorem 3 is not satisfied

[0189] Select parameter σ a =-0.71,σ b =9.75, Select the sequence [t ι ,τ ι ) and [τ ι ,t ι+1 )as follows:

[0190]

[0191] It is easy to verify that condition (41) in Theorem 3 no longer holds. The initial condition is selected as [X, Y, θ] = [-1.3, -2, 0.45], and the values ​​of the other parameters are consistent with those in Case 1. Figure 5 shows the robot's motion, Figure 5 It can be seen that when the condition in Theorem 3 is not satisfied, the controller designed by the present invention will fail. The above simulation results illustrate the effectiveness of the theory proposed by the present invention.

[0192] like Figure 1 The following is a block diagram of a cyber-physical system under a non-periodic denial of service attack. The specific steps for security analysis and control of a cyber-physical system under a non-periodic denial of service attack are as follows:

[0193] When there is no external disturbance:

[0194] Step 1: Construct a physical controlled object model of the cyber-physical system.

[0195] Step 2: Model the non-periodic denial of service attack and then construct a closed-loop system under the non-periodic denial of service attack.

[0196] Step 3: For the closed-loop system under non-periodic denial of service attacks, construct the asymptotic stability conditions for the closed-loop system. In a time-varying attack cycle, derive the relationship between the attack activation interval and the attack silent interval.

[0197] Step 4: Design a secure controller for cyber-physical systems under non-periodic denial-of-service attacks.

[0198] In the presence of external disturbances:

[0199] Step 1: Construct a physical controlled object model of the cyber-physical system with external disturbances.

[0200] Step 2: Construct a closed-loop system under non-periodic denial of service attacks and external disturbances.

[0201] Step 3: For closed-loop systems under non-periodic denial of service attacks and external disturbances, with a guaranteed H ∞ Performance Level Based on this, the asymptotic stability conditions for the closed-loop system are constructed and the relationship between the attack activation interval and the attack silent interval within a time-varying attack cycle is derived.

[0202] Step 4: Design a safety controller so that the closed-loop system is asymptotically stable and has a guaranteed H ∞ Performance Level

[0203] The effectiveness of the method proposed in this invention is verified by using a wheeled mobile robot.

[0204] The above calculation examples of the present invention are only used to explain the calculation model and calculation process of the present invention in detail, and are not intended to limit the implementation methods of the present invention. For ordinary technicians in the relevant field, other different forms of changes or modifications can be made based on the above description. It is impossible to list all the implementation methods here. All obvious changes or modifications derived from the technical solution of the present invention are still within the scope of protection of the present invention.

Claims

1. A security control method for a cyber-physical system under a non-periodic denial of service attack, characterized in that: The method specifically comprises the following steps: Step 1: Build a cyber-physical system model; Step 2: Based on the cyber-physical system model and the aperiodic denial of service attack model, a closed-loop system under aperiodic denial of service attack is constructed; The non-periodic denial of service attack model is as follows: in, It is a non-periodic denial of service attack, k represents the time, using the set represents the set of time-varying periods of aperiodic denial-of-service attacks, t ι <t ι+1 ,ι=1,2,…, is the time-varying interval where there is no denial of service attack, When there is no external disturbance: The cyber-physical system model is as follows: x k+1 =Ax k +Bu k (1) Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, is a real number, m and n are dimensions, matrices A and B are constant matrices; When the attacker does not impose a non-periodic denial of service attack, the designed controller for: Where K represents the controller gain matrix to be designed; Under a non-periodic denial of service attack, the control signal u received by the actuator is k for: Then the closed-loop system under the non-periodic denial of service attack is: In the presence of external disturbances: The cyber-physical system model under external disturbance is: Among them, x k+1 is the system state at time k+1, x k is the system state at time k, u k is the control signal received by the actuator at time k, A and B are constant matrices, C, E and F are constant matrices, ω k represents the external disturbance at time k, z k is the control output at time k, is a real number, m, n, p and q are dimensions; When the attacker does not impose a non-periodic denial of service attack, the designed controller for: Where K represents the controller gain matrix to be designed; Under a non-periodic denial of service attack, the control signal u received by the actuator is k for: When there is a non-periodic denial of service attack, the closed-loop system under external disturbance is: Step 3: Construct the asymptotic stability conditions of the closed-loop system under non-periodic denial of service attacks; When there is no external disturbance, the asymptotic stability condition is: Given a scalar constant -1<σ a <0,σ b >0, if there exists a symmetric matrix P>0 and the gain matrix K satisfies equations (5) to (8), then the closed-loop system of equation (4) is asymptotically stable: (A-BK) T P(A-BK)-P-σ a P<0 (5) A T PA-P-σ b P<0 (6) The superscript T represents the transpose of the matrix, and sup represents t i+1 -τ i The upper bound of , ln is the natural logarithm; When there is an external disturbance, the asymptotic stability condition is: Given a scalar constant -1<σ a <0,σ b >0 and H ∞ Performance Level If there exists a symmetric matrix P>0 and a gain matrix K that satisfies the inequalities of equations (17) to (20), then the closed-loop system of equation (15) is asymptotically stable: Among them, 11 =(A-BK) T P(A-BK)-P-σ a P+C T C,Υ 21 =A T PA-P-σ b P+C T C, the superscript T represents the transpose of the matrix, I is the identity matrix, and sup represents t i+1 -τ i The upper bound of , ln is the natural logarithm; Step 4: Design a controller for the cyber-physical system under a non-periodic denial of service attack to ensure that the closed-loop system is asymptotically stable under a non-periodic denial of service attack. When there is no external disturbance, the design process of the controller of the cyber-physical system is: Solve the matrix X that satisfies inequalities (9) and (10) p and matrix K p , and X p >0: Wherein, the "*" in formula (9) represents (AX p -BK p ) T , the "*" in formula (10) represents (AX p ) T , -1<σ a <0,σ b >0; Then the gain matrix K is: in, is the matrix X p The inverse matrix of Then, according to the gain matrix K, the controller of the cyber-physical system when there is no external disturbance is obtained: When there are external disturbances, the design process of the controller of the cyber-physical system is: Solve the matrix X that satisfies inequalities (21) and (22) p and matrix K p , and X p >0: where, -1 < σ a < 0 and σ b > 0; Then the gain matrix K is designed as: Among them, the superscript -1 represents the inverse of the matrix; Then, according to the gain matrix K, the controller of the cyber-physical system with external disturbance is obtained:

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